Integral Representation by Boundary Vector Measures

1982 ◽  
Vol 25 (2) ◽  
pp. 164-168 ◽  
Author(s):  
Paulette Saab
Keyword(s):  
Banach Space ◽  
Integral Representation ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Linear Subspace ◽  
Vector Measure ◽  
Vector Measures ◽  
Boundary Vector

AbstractIn this paper we show that if X is a compact Hausdorff space, A is an arbitrary linear subspace of C(X, C), and if E is a Banach space, then each element L of (A ⊗ E)* can be represented by a boundary E*-valued vector measure of the same norm as L.

A Choquet theorem for general subspaces of vector-valued functions

1985 ◽  
Vol 98 (2) ◽  
pp. 323-326 ◽  
Author(s):  
Paulette Saab ◽  
Michel Talagrand
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Linear Subspace ◽  
Vector Measure ◽  
Complex Banach Space ◽  
Supremum Norm ◽  
Bounded Linear ◽  
Vector Valued Functions ◽  
Bounded Linear Functional ◽  
Vector Valued

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.

Points of Weak*-Norm Continuity in the Unit Ball of the Space WC(K; X)*

1999 ◽  
Vol 42 (1) ◽  
pp. 118-124 ◽  
Author(s):  
T. S. S. R. K. Rao
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Vector Measures ◽  
Weak Norm ◽  
Norm Continuity ◽  
Nikodym Property ◽  
Isolated Points ◽  
Dense Set

AbstractFor a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak*-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.

On Additive Operators

1971 ◽  
Vol 23 (3) ◽  
pp. 468-480 ◽  
Author(s):  
N. A. Friedman ◽  
A. E. Tong
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Additive Functional ◽  
Representation Theorems ◽  
Image Position ◽  
Vector Valued ◽  
Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

Bourgain Algebras of Douglas Algebras

1992 ◽  
Vol 44 (4) ◽  
pp. 797-804 ◽  
Author(s):  
Pamela Gorkin ◽  
Keiji Izuchi ◽  
Raymond Mortini
Keyword(s):  
Banach Algebra ◽  
Unit Circle ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Linear Subspace ◽  
Closed Subspace ◽  
Closed Subalgebra ◽  
Douglas Algebras

Let A be a Banach algebra and let B be a linear subspace of A. Recall that A has the Dunford Pettis property if whenever ƒn→ 0 weakly in A* and φn → 0 weakly in A* then φn(ƒn) → 0. Bourgain showed that H∞ has the Dunford Pettis property using the theory of ultraproducts. The Dunford Pettis property is related to the notion of Bourgain algebra, denoted Bb, introduced by [6] Cima and Timoney. The algebra Bb is the set of ƒ in A such that if ƒn → 0 weakly in B then dist(ƒƒn, B) —> 0. Bourgain showed [2] that a closed subspace X of C(L)y where L is a compact Hausdorff space, has the Dunford Pettis property if Xb — C(L). Cima and Timoney proved that Bb is a closed subalgebra of A and that if B is an algebra then B⊂Bb. In this paper we study the Bourgain algebra associated with various algebras of functions on the unit circle T.

scholarly journals Exhaustive operators and vector measures

1975 ◽  
Vol 19 (3) ◽  
pp. 291-300 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Vector Measures ◽  
Borel Functions ◽  
Image Position ◽  
Real Topological Vector Space ◽  
Sequence Of Functions

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 andthen Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representationwhere μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

A note on elementary equivalence of C(K) spaces

1987 ◽  
Vol 52 (2) ◽  
pp. 368-373 ◽  
Author(s):  
S. Heinrich ◽  
C. Ward Henson ◽  
L. C. Moore
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Supremum Norm ◽  
Closed Unit Ball ◽  
Elementary Equivalence ◽  
Bounded Number ◽  
Totally Disconnected

In this paper we give a closer analysis of the elementary properties of the Banach spaces C(K), where K is a totally disconnected, compact Hausdorff space, in terms of the Boolean algebra B(K) of clopen subsets of K. In particular we sharpen a result in [4] by showing that if B(K1) and B(K2) satisfy the same sentences with ≤ n alternations of quantifiers, then the same is true of C(K1) and C(K2). As a consequence we show that for each n there exist C(K) spaces which are elementarily equivalent for sentences with ≤ n quantifier alternations, but which are not elementary equivalent in the full sense. Thus the elementary properties of Banach spaces cannot be determined by looking at sentences with a bounded number of quantifier alternations.The notion of elementary equivalence for Banach spaces which is studied here was introduced by the second author [4] and is expressed using the language of positive bounded formulas in a first-order language for Banach spaces. As was shown in [4], two Banach spaces are elementarily equivalent in this sense if and only if they have isometrically isomorphic Banach space ultrapowers (or, equivalently, isometrically isomorphic nonstandard hulls.)We consider Banach spaces over the field of real numbers. If X is such a space, Bx will denote the closed unit ball of X, Bx = {x ϵ X∣ ∣∣x∣∣ ≤ 1}. Given a compact Hausdorff space K, we let C(K) denote the Banach space of all continuous real-valued functions on K, under the supremum norm. We will especially be concerned with such spaces when K is a totally disconnected compact Hausdorff space. In that case B(K) will denote the Boolean algebra of all clopen subsets of K. We adopt the standard notation from model theory and Banach space theory.

scholarly journals STRONGLY BOUNDED REPRESENTING MEASURES AND CONVERGENCE THEOREMS

2010 ◽  
Vol 52 (3) ◽  
pp. 435-445 ◽  
Author(s):  
IOANA GHENCIU ◽  
PAUL LEWIS
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Convergence Theorems

AbstractLet K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

scholarly journals A Remark on the Continuity of the Dual Process

1968 ◽  
Vol 32 ◽  
pp. 287-295 ◽  
Author(s):  
Mamoru Kanda
Keyword(s):  
Banach Space ◽  
Compact Support ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Finite Measure ◽  
Uniform Norm ◽  
Continuous Functions ◽  
Dual Process ◽  
Space Of Continuous Functions ◽  
Locally Compact

Let S be a locally compact (not compact) Hausdorff space satisfying the second axiom of countability and let ℬ be the σ field of all Borel subsets of S and let A be the σ-field of all the subsets of S which, for each finite measure μ defined on (S, A), are in the completed σ field of ℬ relative to μ. We denote by C0 the Banach space of continuous functions vanishing at infinity with the uniform norm and Bk the space of bounded A-measurable functions with compact support in S.

scholarly journals Double-dual types over the Banach spaceC(K)

2005 ◽  
Vol 2005 (16) ◽  
pp. 2533-2545
Author(s):  
Markus Pomper
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Isolated Points

LetKbe a compact Hausdorff space andC(K)the Banach space of all real-valued continuous functions onK, with the sup-norm. Types overC(K)(in the sense of Krivine and Maurey) can be uniquely represented by pairs(ℓ,u)of bounded real-valued functions onK, whereℓis lower semicontinuous,uis upper semicontinuous,ℓ≤u, andℓ(x)=u(x)for all isolated pointsxofK. A condition that characterizes the pairs(ℓ,u)that represent double-dual types overC(K)is given.

scholarly journals Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Manuel Felipe Cerpa-Torres ◽  
Michael A. Rincón-Villamizar
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Geometrical Parameter ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Continuous Image ◽  
Topological Properties ◽  
Locally Compact ◽  
Regular Subspace ◽  
Locally Compact Hausdorff Space

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

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